In the Marvel Universe, a few heroic mantles have been shared by different characters, creating a sense of legacy for the characters involved. The identity of Hawkeye is a prime example of this, with both the original, Clint Barton, and his successor, Kate Bishop, concurrently using the name of the Avenging Archer. However, when the two Hawkeyes first met, they weren't exactly on the best of terms.
Clint and Kate first met in 2007's Fallen Son: The Death of Captain America #3, by Jeph Loeb, John Romita Jr., Klaus Janson, Morry Hollowell, Richard Starkings and Comicraft. Clint met his successor at a time when Iron Man was trying to turn him into the new Captain America. Kate had no idea that Clint had returned from the dead, and Clint wasn't aware of a new Hawkeye either.
Things changed, however, when Clint was tasked with bringing in the Patriot and Bishop's Hawkeye for violating the Superhuman Registration Act. Upon first seeing Kate, Clint was furious that someone stole his old name. The new Captain America confronted Hawkeye about her chosen codename, but Kate masterfully turned the tables on Clint.
Without realizing she was speaking to Clint, Kate said she took the name Hawkeye to honor the original Avenging Archer. If Clint were alive, Kate argued, she would choose a different name. Regardless, Kate would never wear Hawkeye's old costume, a point she used to question Clint wearing Captain America's old uniform. The new Hawkeye made compelling points that ultimately led to Clint giving up the Captain America identity after letting her and the Patriot go free.
The next time the two Hawkeyes met, in 2008's Young Avengers Presents #6, by Matt Fraction, Alan Davis, Mark Farmer, Paul Mounts and Cory Petit, both archers were aware of each other's true identities. By this point in time, Clint had become Ronin of the New Avengers, ambushing Kate in order to test her skills.
Later, Clint invited Kate to the New Avengers headquarters, where the original Hawkeye revealed that he was alive and well. Clint challenged Kate to an archery competition, simultaneously besting the new Hawkeye and giving her advice. As a result of losing the competition, Kate returned Clint's bow.
This was quite humiliating for the young Hawkeye, but eventually, she and her teammate Speed decided to steal the bow back. Clint actually approved of this stunt, impressed by how Kate was willing to stand up for what she believed in. The old Hawkeye saw a little of himself in Kate, from back when he first joined the Avengers. Admitting that Kate and the other Young Avengers were the future, Clint proudly endorsed the new Hawkeye. This endorsement defined Clint and Kate's relationship from this point forward. Even after Clint became Hawkeye again, Kate kept the title, and the two archers have maintained a close relationship ever since.
In 2012, Hawkeye #2, by Matt Fraction, David Aja, Matt Hollingsworth and Chris Eliopoulos saw Clint asking Kate for help in defending his neighborhood. The main reason Clint sought Kate out is because he still saw much of himself in the new Hawkeye. Much like Clint's role on the Avengers, Kate is willing to challenge and question others, which Clint needs sometimes. In this way, Kate carries on the legacy of Hawkeye by calling people out for their faults. Additionally, by acting as a sort of big brother to Kate, Clint has helped her develop as a hero. With Clint's approval and guidance, Kate has become a strong Hawkeye in her own right, even outside of the Young Avengers.
Even though Kate is perfectly capable of fighting her own battles, she still needs help sometimes, and that's why she calls in Clint. The two Hawkeyes have even served on the West Coast Avengers together, a team which Clint helped found. Even though their first couple of meetings were a bit tense, Clint Barton and Kate Bishop eventually found their footing, working together to keep the Hawkeye mantle alive and well.